# The Number Behind the Simplest SIC–POVM

@article{Bengtsson2016TheNB, title={The Number Behind the Simplest SIC–POVM}, author={Ingemar Bengtsson}, journal={Foundations of Physics}, year={2016}, volume={47}, pages={1031-1041} }

The simple concept of a SIC poses a very deep problem in algebraic number theory, as soon as the dimension of Hilbert space exceeds three. A detailed description of the simplest possible example is given.

## 29 Citations

### SICs and Algebraic Number Theory

- Mathematics
- 2017

We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s…

### Sporadic SICs and the Exceptional Lie Algebras

- MathematicsA First Course in the Sporadic SICs
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Sometimes, mathematical oddities crowd in upon one another, and the exceptions to one classification scheme reveal themselves as fellow-travelers with the exceptions to a quite different taxonomy.

### Fibonacci Fervour in Linear Algebra and Quantum Information Theory

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This is a survey on certain results which bring about a connection between Fibonacci sequences on the one hand and the areas of matrix theory and quantum information theory, on the other.

### Moment maps and Galois orbits for SIC-POVMs

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- 2019

The equations that define covariant SIC-POVMs are interpreted in terms of moment maps. Attention is focussed on orbits of a cyclic subgroup of a maximal torus and their images in the moment polytope.…

### Invariant Off-Diagonality: SICs as Equicoherent Quantum States

- Physics
- 2019

Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric…

### Fibonacci-Lucas SIC-POVMs

- Mathematics
- 2017

We present a conjectured family of symmetric informationally complete positive operator valued measures which have an additional symmetry group whose size is growing with the dimension. The symmetry…

### The overlap phases of SIC-POVMs

- MathematicsJournal of Physics: Conference Series
- 2020

A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important…

### Quantum Theory as Symmetry Broken by Vitality

- Physics
- 2019

I summarize a research program that aims to reconstruct quantum theory from a fundamental physical principle that, while a quantum system has no intrinsic hidden variables, it can be understood using…

### Moment Maps and Galois Orbits in Quantum Information Theory

- MathematicsSIAM J. Appl. Algebra Geom.
- 2020

The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which the authors describe explicitly.

### Modular Welch Bounds with Applications

- Mathematics
- 2022

Results (1) and (2) reduce to the famous result of Welch [IEEE Transactions on Information Theory, 1974 ] obtained 48 years ago. We introduce the notions of modular frame potential, modular…

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We give an overview of some remarkable connections between symmetric informationally complete measurements (SIC-POVMs, or SICs) and algebraic number theory, in particular, a connection with Hilbert’s…

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Symmetric informationally complete quantum measurements, or SICs, are mathematically intriguing structures, which in practice have turned out to exhibit even more symmetry than their definition…

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We report on a new computer study into the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are…

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It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

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The Galois group of SICs covariant with respect to the Weyl-Heisenberg group is examined and a list of nine conjectures concerning its structure are proposed, representing a considerable strengthening of the theorems actually proved.

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We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the…

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The existence of maximal sets of equiangular lines (SIC-POVMs) is of interest to the mathematics and physics communities due to their connection to quantum information theory, quantum cryptography,…

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The paper concerns Weyl‐Heisenberg covariant SIC‐POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When…